Testing independence

Test
Independence
Conditional Independence
Simpson
Chi-square
Published

September 2, 2024

Code

Objectives

Confidence Intervals

We start with Confidence Intervals in a simple Gaussian setting. We have \(X_1, \ldots, X_n \sim_{i.i.d.} \mathcal{N}(\mu, \sigma^2)\) where \(\mu\) and \(\sigma\) are unknown (to be estimated and/or tested).

The maximum likelihood estimator for \((\mu, \sigma^2)\) is \((\overline{X}_n, \widehat{\sigma}^2)\) where

\[\overline{X}_n =\sum_{i=1}^n \frac{1}{n} X_i\quad\text{and}\quad \widehat{\sigma}^2=\frac{1}{n}\sum_{i=1}^n (X_i - \overline{X}_n)^2\] By Student’s Theorem \(\overline{X}_n\) and \(\widehat{\sigma}^2\) are stochastically independent \(\overline{X}_n \sim \mathcal{N}(\mu, \widehat{\sigma}^2/n)\) and \(n \widehat{\sigma}^2/\sigma^2 \sim \chi^2_{n-1}\).

\[\sqrt{n} \frac{\overline{X}_n - \mu}{\widehat{\sigma}} \sim t_{n-1}\]

where \(t_{n-1}\) denotes the Student’s \(t\) distribution with \(n-1\) degrees of freedom.

We have the following confidence interval for \(\mu\) at confidence level \(1-\alpha\):

\[\left[\overline{X}_n - \frac{\widehat{\sigma} t_{n-1,\alpha/2}}{\sqrt{n}}, \overline{X}_n + \frac{\widehat{\sigma} t_{n-1,\alpha/2}}{\sqrt{n}}\right]\]

Question

Simulate \(N=1000\) Gaussian samples of size \(n=100\).

Compute the empirical coverage of confidence intervals for \(\alpha=5\%\) and \(\alpha=10\%\).

Plot a histogram for replicates of \(\frac{\overline{X}_n - \mu}{\widehat{\sigma}\sqrt{n}}\). Overlay the density of \(t_{n-1}\).

Testing independence

In data gathered from the 2000 General Social Survey (GSS), one cross classifies gender and political party identification. Respondents indicated whether they identified more strongly with the Democratic or Republican party or as Independents. This is summarized in the next contingency table (taken from Agresti Introduction to Categorical Data Analysis).

Code
# GSS <- vcdExtra::GSS

T <- tribble(~ Democrat, ~ Independent, ~ Republican,
             762, 327, 468,
             484, 239, 477)
rownames(T) <- c('Females', 'Males')
Warning: Setting row names on a tibble is deprecated.
Code
T <- as.matrix(T)
T <- as.table(T)
names(dimnames(T)) <- c("Gender", "Party identification")
Code 
         Party identification
Gender      Democrat Independent Republican
  Females 0.27638738  0.11860718 0.16974973
  Males   0.17555314  0.08668843 0.17301415
Code 
Gender
Females   Males 
   1557    1200
Code 
Party identification
   Democrat Independent  Republican 
       1246         566         945
Question
  • Draw mosaicplot for the cross classification table
  • Compute the Pearson chi-square statistic for testing independence
  • Comment

Visualizing multiway categorical data

Consider the celebrated UCBAdmissions dataset

According to R documentation, this dataset is made of

Aggregate data on applicants to graduate school at Berkeley for the six largest departments in 1973 classified by admission and sex.

This is a compilation of 4526 application files.

For each application, three variables have been reported: the department , the gender of the applicant, and whether the applicant has been admitted.

The dataset is a trivariate sample, which is summarized by a 3-way contingency table.

Code
data("UCBAdmissions")
Question

Turn the 3-way contingency table into a dataframe/tibble with columns Gender, Dept, Admit, n, where the first columns are categorical, and the last column counts the number of co-occurrences of the values in the first three columns amongst the UCB applicants.

Question

Make it a bivariate sample by focusing on Gender and Admit: compute the margin table

Draw the corresponding mosaicplot and compute the chi-square independence statistic.

Comment.

Question

Visualize the three-way contingency table using double-decker plots from vcd

Question
Question

Viewing the UCBAdmissions dataset, which variable would you call a response variable? Which variable would you call covariates?

Test independence between Gender and Dept.

Question

For each department of application (Dept), extract the partial two-way table for Gender and Admit. Test each two-way table for independence. How many departments pass the test at significance level \(1\%\), \(5\%\)?

Note that the two-way cross-sectional slices of the three-way table are called partial tables.

What we observed has a name.

Simpson’s paradox

The result that a marginal association can have different direction from the conditional associations is called Simpson’s paradox. This result applies to quantitative as well as categorical variables.